SUNA37, Ideal Flow addon's
===========================
This article is kind of supplement to my postings about the Labrujere Problem,
which is the Least Squares Finite Element approximation of Ideal Flow around a
circular cylinder in velocity (u,v) formulation (: the Cauchy equations).
The series is accompanied with a couple of computer programs in GWBASIC. It was
decided to implement another series of numerical experiments, in order to make
the overall picture more complete. It was attempted to calculate velocities at
the vertices of finite elements, instead of the midside nodes.
The abovementioned GWBASIC programs have been made publically available via the
well known anonymous ftp site at "dutrde1.tudelft.nl".
"CURRENT.DAT" contains a coding for the (Ideal Flow) problem to be calculated,
a two digit number X . At the moment of this writing the following 6 problems
have been implemented in addition to existing cases (SUNA30):
X = 04 : like the original mesh devised by Labrujere
X = 05 : four triangles in a quadrilateral
X = 06 : same arrangement, but only even numbered elements are counted
X = 07 : my original reduced quadrilaterals
X = 08 : reduced quadrilaterals with linearization
X = 09 : same arrangement, but only even numbered elements are counted
Before running the main program, data must be created. This is done by invoking
a mesh generator: "GWBASIC NLRGEN". Direct access files are created then, such
as: GENER04.BIN , COORD04.BIN , TOPOL04.BIN , BOUND04.BIN .
The six Least Squares Finite Elements types, all with velocities defined at the
vertices, are implemented in a program called NLRFLOW.BAS. Invoking "NLRFLOW"
carries out the main flow calculation. As it ends, a direct access file like
VELOC04.BIN, containing the outcome, is created.
The mesh, together with the calculated velocities, can be plotted on a graphics
(true or simulated CGA) screen by typing "GWBASIC NLRPLOT".
The quality of the flow velocity calculation can be judged by comparing it with
the analytical solution, which is found by a complex Joukowski transformation.
Especially the velocity profile at the cylinder itself is significant (because
the circular cylinder is in fact kinda prototype for the wing of an airplane).
A program called NLRESULT is invoked to carry out the comparison between the
above analytical and our numerical approximations. The results are represented
graphically, as in the original NLR report by Labrujere. The outcome is:
X = 04 : bad, like in the original report by Labrujere
X = 05 : almost equally bad
X = 06 : still bad or even worse (unlike the SUNA29 case)
X = 07 : the only _good_ result, despite of wrong elements
X = 08 : comparable with case 05, as expected from theory
X = 09 : comparable with case 06, as expected from theory
Especially the bad results for X = 06 and X = 09 deserve some explanation.
At first sight, the elements used are the same "Linear Quadrilaterals" as have
been _succesfully_ applied via SUNA29 in MAINFLOW.BAS, the obvious difference
being a rotation over 45 degrees. But if we look more closely, there is more.
If we join midpoints at the sides of an arbitrary quadrilateral, then always a
paralellogram results. Coordinates of such a paralellogram obey the following
relationship:
x1 + x4 = x2 + x3 (due to the fact that the diagonals
3  4 y1 + y4 = y2 + y3 divide each other in equal pieces)
/ /
/ / Because of Isoparametric Transformations to be applied in
1  2 a consequent fashion, it is "natural" to assume that this
linearity holds in general: f1 + f4 = f2 + f3 .
No such heuristics can be devised for linearization, when applied directly to
reduced quadrilaterals with velocities at the vertex nodes. These elements are
in general NO paralellograms. Hence, any linearity for x and y is absent.
Why then expect that linearity still can hold for quantities like u and v ?
Another possible reason for failure is that defining proper boundary conditions
turns out to be quite a bit different for vertex and midside node elements.
In addition to our flow calculation programs, two BASICs for testing the theory
in SUNA33 have been developed. They are called "temper.bas" and "nlrtemp.bas".
The 1st utilizes midside velocities (case X = 01,02,03 ) while the 2nd utilizes
vertex velocities (case X = 07). Thus the Ideal Flow field around a circular
cylinder has been used here as a first testcase for FLOTRANSUFED unification.
The temperature field can be visualized by plotting isotherms. This is done by
a slightly modified version of the "psiplot" program, called now "conplot.bas".
To be continued ...

* Han de Bruijn; Applications&Graphics  "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354  would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands.  Mathematics" (HdB). * for
* Email: Han.deBruijn@RC.TUDelft.NL  Fax: +31 15 78 37 87 * Blood